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Cayley diagram

C2×Dic25 #6020

order200
isoclinismisoclinic class of D25
automorphismAut(G) = D4×AGL(1,ℤ/25) (order 4000) · Inn(G) = D25 · Out(G) = C2×C5×D4
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2

Aliases

direct productC2×Dic25

Splittings

derived, SylowC25 ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC22D25
commutator, hypocenterC25C2×C4
FrattiniC2×C5C2×D5
FittingC22×C25C2
socle, prime radicalC22×C5D5
cosocleC2×C25C22
LCSG ▷ C25
UCSC1 ◁ C22

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 200

A minimal faithful permutation representation — 2 generators on 31 points:

(1,2)(3,4,5,6)(8,10)(9,13)(11,16)(12,19)(14,22)(15,21)(17,18)(20,26)(23,27)(24,28)(25,30)(29,31);(1,2)(7,21,31,24,10,13,27,25,16,19,22,20,18,17,26,14,12,11,30,23,9,8,28,29,15)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×Dic25, #87718, C4⋊Dic25, …
maximal normal subgroup ofC4×Dic25, #87718, C4⋊Dic25, …
maximal quotient ofC2×#6014, #87716, C4×Dic25, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(200, 7);